Yes, what @yrtimiD said is correct.
The names of bus stops on OSM should match the name that appears on the sign. This name is also what appears in navigation apps (Egg, Rav-Pass, Moovit, …) and official websites like egged.co.il. If you want the name changed, you might have some luck contacting the Ministry of Transportation and letting them know.
If you can show me a photo that the physical sign on the stop doesn’t have the גרשיים, then I might look into changing the script’s policy. But until then, I have to assume the data from the MoT is accurate. Edit: and even in that case, we should just inform the MoT to update the data in the GTFS feed. I can contact them, they usually respond. Not always, though.
I’ll stick with English simply because you already did, but I’m equally comfortable with both languages so feel free to switch if you’re better with Hebrew.
Not that I know of! That would be nice, though…
This is deliberate. Nice that you noticed! 
I did my best to document it in the wiki in the Name section:
The name is improved from the GTFS name according to these rules:
- Repeated single-quotation marks
'' are converted to a double-quotation mark ". The GTFS data likely uses '' to avoid double-quotes being interpreted as delimiting a substring in the CSV format.
- If there is a leading single-quotation mark, then it’s supposed to be a Geresh and it’s moved to the end. A lot of stops have this character in the wrong place.
- Non-breaking space is replaced with normal space.
- Leading and trailing whitespace is removed, and repeated spaces are merged together.
- Double-quotation marks surrounded by Hebrew letters are converted to Gershayim (U+05F4 ״).
- Single-quotation marks preceded by a Hebrew letter are converted to Geresh (U+05F3 ׳).
And in the later GTFS tags section:
gtfs:stop_name:IL-MOT=* - the Hebrew stop name, exactly as it appears in the GTFS data, without the improvements detailed above, except that leading and trailing whitespace is removed.
I hope you’ll agree that this results in better formatting. I am strongly in favor of using U+05F4 גרשיים instead of U+0022 quotation mark. Quotation mark can sometimes break text rendering in systems that don’t expect it to appear in the middle of a word. And I certainly won’t stand for two apostrophes, which is what the Ministry of Transportation provides!